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We consider Malliavin calculus based on the It\^o chaos decomposition ofsquare integrable random variables on the L\'evy space. We show that when arandom variable satisfies a certain measurability condition, itsdifferentiability and fractional differentiability can be determined byweighted Lebesgue spaces. The measurability condition is satisfied for allrandom variables if the underlying L\'evy process is a compound Poisson processon a finite time interval.

A note on Malliavin smoothness on the L\'evy space